Duplicate-Aware Shift-and-Lift Carleman Linearization:Structure, Complexity, and Comparative Evaluation
For researchers using Carleman linearization in nonlinear dynamics, this method reduces computational overhead and improves local convergence, though gains are incremental and context-dependent.
This work introduces a duplicate-aware shift-and-lift Carleman linearization that removes redundant monomials, enabling high-order truncation with reduced complexity. Benchmarks show regime-dependent accuracy gains over Jacobian linearization, but no universal superiority.
The primary objective of this study is to remove duplicated monomial contributions that proliferate in Carleman linearization as state dimension and truncation order increase. To do so, we adopt a shift-and-lift architecture, since it exposes repeated exponent targets and allows duplicate-aware coefficient coalescing during lifted-operator assembly. This architecture also makes high-order truncation practical, but that regime intensifies local convergence and closure sensitivity for higher-order nonlinearities. We therefore pair shift-and-lift with a moving-center expansion so that shift and lift are updated jointly around evolving local centers, improving validity of the truncated model along the trajectory. The resulting workflow combines symmetry-reduced monomial bases, packed exponent-key indexing, and sparse triplet coalescing to preserve truncated affine dynamics while reducing index-resolution overhead and write-path irregularity. We analyze variable growth, preprocessing complexity, and truncation-induced error mechanisms, and we compare against Jacobian linearization through fixed-step error, admissible step size, and cost-at-target-accuracy criteria. Two benchmarks (bilinear driver and logistic interaction) show convergence under refinement for both approaches, with regime-dependent accuracy gains for the proposed method rather than universal superiority.